Portfolio selection problems in practice: a comparison between linear and quadratic optimization models

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model, based on reformulation as a Standard Quadratic Program and on some recent theoretical results. With this approach we obtain optimal solutions both for some well-known financial data sets used by several other authors, and for some unsolved large size portfolio problems. We also test our method on five new data sets involving real-world capital market indices from major stock markets. Our computational experience shows that, rather unexpectedly, it is easier to solve the quadratic LAM model with our algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of the best commercial codes for mixed integer linear programming (MILP) problems. Finally, on the new data sets we have also compared, using out-of-sample analysis, the performance of the portfolios obtained by the Limited Asset models with the performance provided by the unconstrained models and with that of the official capital market indices

Portfolio selection problems in practice: a comparison between linear and quadratic optimization models

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model, based on reformulation as a Standard Quadratic Program and on some recent theoretical results. With this approach we obtain optimal solutions both for some well-known financial data sets used by several other authors, and for some unsolved large size portfolio problems. We also test our method on five new data sets involving real-world capital market indices from major stock markets. Our computational experience shows that, rather unexpectedly, it is easier to solve the quadratic LAM model with our algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of the best commercial codes for mixed integer linear programming (MILP) problems. Finally, on the new data sets we have also compared, using out-of-sample analysis, the performance of the portfolios obtained by the Limited Asset models with the performance provided by the unconstrained models and with that of the official capital market indices

Portfolio selection based on minmax rule and fuzzy set theory.

Yang, Fan.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (p. 100-106).Abstracts in English and Chinese.Abstract --- p.iAcknowledgement --- p.iiiChapter 1 --- Introduction --- p.1Chapter 1.1 --- Literature review --- p.1Chapter 1.2 --- The main contribution of this thesis --- p.5Chapter 1.3 --- Relations between the above three models --- p.7Chapter 2 --- Model 1 --- p.9Chapter 2.1 --- Introduction --- p.9Chapter 2.2 --- Minimax rule risk function --- p.11Chapter 2.3 --- Fuzzy liquidity of asset --- p.12Chapter 2.4 --- Notations --- p.15Chapter 2.5 --- Model formulation --- p.16Chapter 2.6 --- Numerical example and result --- p.25Chapter 3 --- Model 2 --- p.36Chapter 3.1 --- Introduction --- p.36Chapter 3.2 --- Notations --- p.39Chapter 3.3 --- Model formulation --- p.41Chapter 3.4 --- Numerical example and result --- p.45Chapter 4 --- Model 3 --- p.51Chapter 4.1 --- Introduction --- p.51Chapter 4.2 --- Notations --- p.52Chapter 4.3 --- Model formulation --- p.54Chapter 4.4 --- Numerical example and result --- p.62Chapter 5 --- Conclusion --- p.68Chapter A --- Source Data for Model 1 --- p.71Chapter B --- Source Data for Model 2 --- p.80Chapter C --- Source Data for Model 3 --- p.90Bibliography --- p.10

Portfolio optimization under minimax risk measure with investment bounds.

Wong, Chi Ying.Thesis (M.Phil.)--Chinese University of Hong Kong, 2007.Includes bibliographical references (leaves 71-74).Abstracts in English and Chinese.Abstract Page --- p.iiAcknowledgment Page --- p.ivChapter 1 --- Introduction --- p.1Chapter 2 --- Literature Review --- p.5Chapter 3 --- Review of minimax portfolio selection model --- p.11Chapter 3.1 --- The I∞ model --- p.11Chapter 4 --- Portfolio optimization with group investment limits --- p.16Chapter 4.1 --- The model --- p.16Chapter 4.2 --- The optimal investment strategy --- p.17Chapter 4.2.1 --- All assets are risky --- p.18Chapter 4.2.2 --- Some riskfree assets are involved --- p.39Chapter 4.3 --- Chapter summary --- p.40Chapter 5 --- Tracing out the efficient frontier --- p.41Chapter 5.1 --- Properties of the efficient frontier --- p.42Chapter 5.2 --- The algorithm --- p.51Chapter 5.3 --- Time complexity of the algorithm --- p.56Chapter 5.4 --- Chapter summary --- p.57Chapter 6 --- Finding the investor's optimal portfolio --- p.58Chapter 6.1 --- Investor's portfolio with given A --- p.58Chapter 6.2 --- Chapter summary --- p.60Chapter 7 --- Numerical experiments --- p.61Chapter 7.1 --- Finding the efficient frontier numerically --- p.61Chapter 7.2 --- Performance between mean-variance model and I∞ model --- p.64Chapter 7.2.1 --- Data analysis --- p.64Chapter 7.2.2 --- Experiment description and discussion --- p.65Chapter 7.3 --- Chapter summary --- p.67Chapter 8 --- Conclusion --- p.68Bibliography --- p.71Appendix --- p.75Chapter A --- Stocks for finding the efficient frontiers with and without bound constraints --- p.75Chapter B --- List of companies --- p.77Chapter C --- Graphical Results --- p.8

A multi-period portfolio selection problem.

Hou, Wenting.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (p. 113-117).Abstract also in Chinese.Abstract --- p.iAcknowledgement --- p.iiiChapter 1 --- Introduction --- p.1Chapter 1.1 --- Literature Review --- p.1Chapter 1.2 --- Problem Description --- p.8Chapter 1.3 --- The Main Contributions of This Thesis --- p.11Chapter 2 --- Model I --- p.13Chapter 2.1 --- Notation --- p.13Chapter 2.2 --- Model Formulation --- p.16Chapter 2.3 --- Analytical Solution --- p.19Chapter 3 --- Model II --- p.25Chapter 3.1 --- Model Formulation --- p.25Chapter 3.2 --- Analytical Solution --- p.30Chapter 3.3 --- How to Find y --- p.38Chapter 3.4 --- Numerical Example --- p.42Chapter 4 --- Model III --- p.47Chapter 4.1 --- Model Formulation --- p.48Chapter 4.2 --- Dynamic Programming --- p.50Chapter 4.2.1 --- DP I --- p.50Chapter 4.2.2 --- DP II --- p.53Chapter 4.3 --- Approximate Analytical Solution --- p.56Chapter 4.4 --- Computational Result Comparison --- p.65Chapter 5 --- Conclusions --- p.73Chapter A --- Source Data --- p.76Chapter A.l --- rti --- p.76Chapter A.2 --- qti --- p.79Chapter B --- Model II Numerical Example and Result --- p.82Chapter B. --- l Value of xti when A = 0.3 --- p.82Chapter B.2 --- Value of xti when A = 0.6 --- p.84Chapter B.3 --- Value of xti when A = 0.9 --- p.88Chapter B.4 --- True Value of xti --- p.91Chapter C --- Model III Numerical Example and Result --- p.98Chapter C.l --- The Value of Mt of DP II --- p.98Chapter C.2 --- Track of Optimal Value of DP II --- p.101Chapter C.3 --- The Optimal Total Wealth of DP II --- p.105Chapter C.4 --- The Optimal Asset Allocation of P4 --- p.109Bibliography --- p.11

Problems in Mathematical Finance Related to Transaction Costs and Model Uncertainty.

This thesis is devoted to the study of three problems in mathematical finance which involve either transaction costs or model uncertainty or both. In Chapter II, we investigate the Fundamental Theorem of Asset Pricing (FTAP) under both transaction costs and model uncertainty, where model uncertainty is described by a family of probability measures, possibly non-dominated. We first show that the recent results on the FTAP and the super-hedging theorem in the context of model uncertainty can be extended to the case where only options available for static hedging (hedging options) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of robust no-arbitrage which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are non-redundant. Next, we look at the more difficult case where the market consists of a money market and a dynamically traded stock with bid-ask spread. Under a continuity assumption, we prove using a backward-forward scheme that no-arbitrage is equivalent to the existence of a suitable family of consistent price systems. In Chapter III, we study the problem where an individual targets at a given consumption rate, invests in a risky financial market, and seeks to minimize the probability of lifetime ruin under drift uncertainty. Using stochastic control, we characterize the value function as the unique classical solution of an associated Hamilton-Jacobi-Bellman (HJB) equation, obtain feedback forms for the optimal investment and drift distortion, and discuss their dependence on various model parameters. In analyzing the HJB equation, we establish the existence and uniqueness of viscosity solution using Perron's method, and then upgrade regularity by working with an equivalent convex problem obtained via the Cole-Hopf transformation. In Chapter IV, we adapt stochastic Perron's method to the lifetime ruin problem under proportional transaction costs which can be formulated as a singular stochastic control problem. Without relying on the Dynamic Programming Principle, we characterize the value function as the unique viscosity solution of an associated variational inequality. We also provide a complete proof of the comparison principle which is the main assumption of stochastic Perron's method.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111560/1/yuchong_1.pd

변동성 위험과 신뢰도 위험을 고려한 최적 전원구성 도출 방법론 연구

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 협동과정 기술경영·경제·정책전공, 2020. 8. 이종수.Long-term power planning has been focused primarily on cost minimization, which was the same in other countries as in Korea. Since 2000, several studies applied Markowitz's portfolio theory to the portfolio of power generation sources. However, many of the earlier studies only concentrated on finding the efficient frontier of the portfolio, and there has not been a study on the trade-off ratio value between the cost and its volatility. Therefore, in earlier studies, the optimal portfolios from the efficiency frontier were found through scenario analysis, and not the real value of the policymaker's trade-off ratio. The primary aim of this paper is to estimate reasonably the exchange ratio between costs and their volatility in the analysis of the optimal power mix using the mean-variance model. This study started from the microeconomic foundation, which the policy makers used to establish the power plan to maximize their social welfare, estimate the marginal rate of substitution (MRS) between these elements using the time series of the power structure in Korea, and derive the optimal power portfolio from this. The secondary aim of this paper is to include in the analysis model the reliability risks that must be considered in the optimal power generation mix. Several studies describe power generation assets in the same way as securities traded in the capital market, but it is very important to maintain power supply reliability as well as minimize cost, and avoid volatility in real-world power plant investment. In this study, the reliability risk was defined as the loss of load probability, and the mean-variance portfolio model was expanded by including it as an element of the social welfare function of policy-makers in establishing a power plan. The findings of the study are as follows: First, from the perspective of cost and volatility, the ratio of substitution between the two factors gradually changed from 1992 to 2014 to take more volatility risk. This was a major reason for the expansion of combined cycle gas turbine, which was eco-friendly and continuously improved in thermal efficiency since the 1990s, whereas diversifying power sources with nuclear power and coal after the oil shock in the 1970s. Second, the actual power generation portfolio was gradually approaching the optimal portfolio during the analysis period, but the share of LNG combined cycle power generation has increased significantly compared to the optimum level since 2011 when a large-scale power outage occurred in Korea. This can be attributed to the fact that in the early 2010s, the approval for the construction of LNG combined cycle power plants increased significantly to cope with the electricity crisis because of a short construction time. Third, when considering power reliability, the ratio of the optimal power generation portfolio was found to increase in proportion to peak-load generator, especially LNG, as compared to the volatility-risk-only model. This is because the combined power generation technology is composed of several gas turbines and a steam turbine, and the unit capacity per generator is small, which has a considerable diversification effect even in the event of generator failure. Based on these results, it is expected that the proportion of LNG in the power generation portfolio will have to be increased in the future. This is because policy makers are gradually changing the viewpoint of allowing volatility risk in their utility, and LNG CC is superior to other power sources in terms of reliability. In particular, the expansion of renewable power sources, which will increase the risk of reliability, is expected to require more LNG facilities in the future.지금까지 장기 전원계획은 주로 비용최소화를 바탕으로 이루어져왔다. 하지만, 2000년대 이후부터 Markowitz의 포트폴리오 이론을 발전설비의 포트폴리오에 적용하는 연구가 본격적으로 이루어지기 시작하면서 큰 변화가 나타났다. 그러나 선행의 많은 연구들은 발전비용의 평균과 분산을 통해 포트폴리오의 효율 경계를 찾는데 주된 목적을 두었고, 그 두 요소 간의 교환비율이 어떻게 되는지에 대한 연구는 이루어지지 않았다. 그래서 효율경계로부터 최적 전원구성의 찾아내는 방법은 시나리오 기법에 의존하거나, 전통적인 CAPM 모형을 이용하여 시장 포트폴리오를 도출하는데 그쳤다. 본 논문의 첫 번째 목적은 평균-분산 모형을 적용한 최적 전원 믹스를 분석함에 있어서, 비용의 평균과 그 변동성 간의 교환 비율, 즉 trade-off 관계를 합리적으로 추정하는데 있다. 두 번째 목적은 최적 전원구성을 고려함에 있어서, 전력산업에서 반드시 고려해야하는 신뢰도 위험을 분석 모형에 반영하는 것이다. 기존의 많은 연구들은 발전 자산이 마치 자본시장에서 거래되는 유가증권과 같은 방식으로 분석되었으나, 현실의 발전설비 투자는 비용최소화와 변동성 회피뿐만 아니라, 전력 신뢰도를 유지하는 것이 매우 중요하다. 본 논문에서는 신뢰도 위험을 공급지장확률(LOLP)로 정의하여, 전원계획을 수립하는 정책당국자의 효용함수의 한 요소로 반영하여 평균-분산 포트폴리오 모형을 확장시켰다. 모형의 미시적 기초는 변동성 위험만을 고려한 1위험 모형과 동일하며, 우리나라의 LOLP함수를 산출하기 위하여 몬테카를로 시뮬레이션을 이용하였다. 이러한 연구목표와 방법론으로부터 얻은 결과는 다음과 같다. 첫째, 비용과 비용의 변동성의 관점에서 정책입안자가 바라보는 두 요소간의 대체 비율은 1992~2014년 동안 점차 변동성을 허용하는 쪽으로 선호가 변경되었다. 이는1970년대 오일쇼크 이후 원자력과 석탄으로 발전원의 다각화를 시도하였다가, 1990년대 이후부터 친환경적이고 발전효율이 지속적으로 개선된 LNG 복합발전이 확대된데 큰 이유가 있었다. 둘째, 실제 전원구성은 분석기간 동안 점차 최적 포트폴리오에 근접해지고 있었으나, 대규모 순환정전이 발생하였던 2011년 이후로 LNG 복합 발전의 비중이 최적에 비해 훨씬 늘어났다. 이는 2010년대 초, 전력 수급위기에 대응하여 건설 기간이 짧은 LNG 복합발전의 건설 승인이 상당수 늘어난데 그 원인을 찾을 수 있다. 셋째, 전력신뢰도를 고려할 경우 최적 전원구성 비율은 변동성만 고려한 모형보다 피크발전설비, 그 중에서도 특히 LNG의 비중이 늘어나는 것으로 나타났다. 이는 복합발전 기술이 여러 대의 가스 터빈과 스팀터빈으로 이루어져, 발전기당 단위 기 용량이 작아 고장 발생에도 상당한 분산 효과가 있기 때문이다. 이러한 결과를 바탕으로 전원구성에의 정책적 시사점을 도출하면, 향후 전원구성에는 현재보다 LNG의 비중이 더 늘어나야 할 것으로 보인다. 이는 정책입안자의 효용도 비용의 변동성을 점차 허용하는 관점으로 변하고 있고, 신뢰도 측면에서도 다른 전원에 비하여 우월한 특성이 있기 때문이다. 특히, 온실가스 배출 비용의 증가와 신뢰도 위험을 증가시킬 신재생 전원의 정책적 확대는 앞으로 더 많은 LNG설비를 필요로 할 것으로 예상된다.Chapter 1. Introduction 1 1.1 Research Background 1 1.2 Research Objectives 4 1.3 Research Outline 6 Chapter 2. Literature Review 7 2.1 Portfolio Theory 7 2.1.1 Markowitzs Concept 8 2.1.2 Capital Asset Pricing Model 10 2.2 Application to Power Generation Mix 14 2.2.1 Application to Global Case 14 2.2.2 Application to Korean Case 19 2.3 Estimation of the Trade-off Ratio 23 2.4 Limitations of Previous Research and Research Motivation 25 Chapter 3. Methodology 29 3.1 Volatility Risk Only Model (1-risk model) 29 3.1.1 Microeconomic Foundation 29 3.1.2 Econometric Method 35 3.2 Reliability Risk Added Model (2-risk model) 40 3.2.1 Measure of Reliability risk 40 3.2.2 Microeconomic Foundation 45 Chapter 4. Empirical Studies 56 4.1 Data Specification 56 4.1.1 Investment Cost 56 4.1.2 O&M and Fuel cost 59 4.1.3 Total Supply Cost 61 4.2 Estimation of 1-risk Model 63 4.2.1 Estimation of Covariance Matrix 63 4.2.2 Estimation of Share Equation 69 4.2.3 Empirical Results and Discussion 70 4.3 Estimation of 2-risk Model 79 4.3.1 Calculation of LOLP 79 4.3.2 Estimation of Share Equation 84 4.3.3 Empirical Results and Discussion 86 4.4 Implication for Electric Power Industry Policy 94 4.4.1 Revisit to the CAPM 95 4.4.2 Intermittency of Renewable Energy 102 4.4.3 Future Portfolio Including Renewable Energy 107 Chapter 5. Summary and Conclusion 111 5.1 Concluding Remarks and Contribution 111 5.2 Limitation and Future Studies 115 Bibliography 116 Appendix 1 : Deriving Optimal Share Equation 128 Appendix 2 : Deriving Derivatives of LOLP Function 130 Appendix 3 : Data Set 133 Appendix 4 : 8th Basic plan for supply and demand 135 Abstract (Korean) 139Docto